It is a non-starter. You can't throw bodies at this problem. The problem needs a real fix. You have imagineered a process that may not possible on their systems. The people you would task to do this can only do things that their computers have been programmed to allow them to do. I don't think it is ten times more complex than you imagine. It is probably a thousand times more complex than you imagine.
(Plus your solution still doesn't solve the OP problem when two different people buy the same item online)
I know there's a ton of websites already (even some mom and pops shops in all types of retail) that handle the issue of 2 people buying the same item online... Once something is sold online, it immediately changes the item to "no longer available/out of stock", so I don't see that being an issue at all really. Auto parts retailers have had that tech for well over a decade now. I can't see why GC couldn't adopt 15 year old tech for that particular issue. Darren Riley (a very small in comparison to GC) online music parts store has that tech on their site. I know becasue I've wanted to buy multiples of the same item from them and it stopped me becasue I bought the last one already, and it was instant.
As far as the issue of having multiple retail stores and one comprehensive website all interacting together, I don't believe it's 1000xs more complex than I imagine as you say... or even 10xs. I say that because with all the thousands of different types of retailers in the world (outside of the musical instrument biz) I can't imagine that people haven't already seen a need for that type software, and that it doesn't already exist in several forms to choose from already. I'm not saying it's cheap, but I don't believe GC would have to pay millions and millions to have an IT company custom make it for them. Just buy what's already been invented. Carmax comes to mind as a first thought.